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Prefixed curves in moduli space
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Buff, Xavier, Epstein, Adam L. and Koch, Sarah (2022) Prefixed curves in moduli space. American Journal of Mathematics, 144 (6). pp. 1485-1509. doi:10.1353/ajm.2022.0036 ISSN 0002-9327.
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Official URL: http://dx.doi.org/10.1353/ajm.2022.0036
Abstract
We study the geometry of certain algebraic curves in the moduli space of cubic polynomials, and in the moduli space of quadratic rational maps. Given $k\geq 0$, ($k\neq 1$ in the case of quadratic rational maps), we show that the set of conjugacy classes of maps with a prefixed critical point of preperiod $k$, is an algebraic curve that is irreducible (over $\Bbb{C}$). We then study a closely related question concerning the irreducibility (over $\Bbb{Q}$) of the set of conjugacy classes of unicritical polynomials, of degree $D\geq 2$, with a preperiodic critical point. Our proofs are purely arithmetic; they rely on a result providing sufficient conditions under which irreducibility over $\Bbb{C}$ is equivalent to irreducibility over $\Bbb{Q}$, and on a generalized Eisenstein criterion for irreducibility.
Item Type: | Journal Article | ||||
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Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||
Journal or Publication Title: | American Journal of Mathematics | ||||
Publisher: | The Johns Hopkins University Press | ||||
ISSN: | 0002-9327 | ||||
Official Date: | 6 December 2022 | ||||
Dates: |
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Volume: | 144 | ||||
Number: | 6 | ||||
Page Range: | pp. 1485-1509 | ||||
DOI: | 10.1353/ajm.2022.0036 | ||||
Status: | Peer Reviewed | ||||
Publication Status: | Published | ||||
Access rights to Published version: | Restricted or Subscription Access |
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